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arXiv:1005.1704 · Physical Review B

Coefficients of bosonized dimer operators in spin- 12 XXZ chains and their applications

Shintaro Takayoshi, Masahiro Sato

Published Dec 16, 2010• 38 citations
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Abstract

Comparing numerically evaluated excitation gaps of dimerized spin-$\frac{1}{2}$ $XXZ$ chains with the gap formula for the low-energy effective sine-Gordon theory, we determine coefficients ${d}_{xy}$ and ${d}_{z}$ of bosonized dimerization operators in spin-$\frac{1}{2}$ $XXZ$ chains, which are defined as ${(\ensuremath{-}1)}^{j}({S}_{j}^{x}{S}_{j+1}^{x}+{S}_{j}^{y}{S}_{j+1}^{y})={d}_{xy}\text{ }\text{sin}[\sqrt{4\ensuremath{\pi}}\ensuremath{\phi}(x)]+\ensuremath{\cdots}$ and ${(\ensuremath{-}1)}^{j}{S}_{j}^{z}{S}_{j+1}^{z}={d}_{z}\text{ }\text{sin}[\sqrt{4\ensuremath{\pi}}\ensuremath{\phi}(x)]+\ensuremath{\cdots}$. We also calculate the coefficients of both spin and dimer operators for the spin-$\frac{1}{2}$ Heisenberg antiferromagnetic chain with a nearest-neighbor coupling $J$ and a next-nearest-neighbor coupling ${J}_{2}=0.2411J$. As applications of these coefficients, we present ground-state phase diagrams of dimerized spin chains in a magnetic field and antiferromagnetic spin ladders with a four-spin interaction. The optical conductivity and electric polarization of one-dimensional Mott insulators with Peierls instability are also evaluated quantitatively.

DOI
10.1103/physrevb.82.214420

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The ground-state phase diagram of a spin-12\frac{1}{2}21​ XXZ chain with competing ferromagnetic nearest-neighbor (J1<0{J}_{1}<0J1​<0) and antiferromagnetic second-neighbor (J2>0{J}_{2}>0J2​>0) exchange couplings is studied by means of the infinite time evolving block decimation algorithm and effective field theories. For the SU(2)-symmetric (Heisenberg) case, we show that the nonmagnetic phase in the range $\ensuremath{-}4<{J}_{1}/{J}_{2}<0$ has a small but finite ferromagnetic dimer order. We argue that this spontaneous dimer order is associated with effective spin-1 degrees of freedom on dimerized bonds, which collectively form a valence bond solid state as in the spin-1 antiferromagnetic Heisenberg chain (the Haldane spin chain). We thus call this phase the Haldane dimer phase. With easy-plane anisotropy, the model exhibits a variety of phases including the vector chiral phase with gapless excitations and the even-parity dimer and N\'eel phases with gapped excitations, in addition to the Haldane dimer phase. Furthermore, we show the existence of gapped phases with coexisting orders in narrow regions that intervene between the gapless chiral phase and any one of Haldane dimer, even-parity dimer, and N\'eel phases. Possible implications for quasi-one-dimensional edge-sharing cuprates are discussed.
We study the phase diagram, ground-state properties, and excitation gaps of a frustrated spin-Peierls chain, i.e., a spin-Peierls chain with both nearest-neighbor exchange J1{\mathit{J}}_{1}J1​ and next-nearest-neighbor exchange J2{\mathit{J}}_{2}J2​. The phase diagram is calculated using a bosonization renormalization-group method. We propose a bond-operator mean-field approximation to calculate the ground-state properties and excitation gaps.
We study the finite-size behavior of the low-lying excitations of spin-12\frac{1}{2}21​ Heisenberg chains with dimerization and next-to-nearest neighbors interaction J2{J}_{2}J2​. The numerical analysis, performed using density-matrix renormalization group, confirms previous exact diagonalization results and shows that, for different values of the dimerization parameter $\ensuremath{\delta}$, the elementary triplet and singlet excitations present a clear scaling behavior in a wide range of $l=L∕\ensuremath{\xi}$ (where LLL is the length of the chain and $\ensuremath{\xi}$ is the correlation length). At J2=J2c{J}_{2}={J}_{2c}J2​=J2c​, where no logarithmic corrections are present, we compare the numerical results with finite-size predictions for the sine-Gordon model obtained using L\"uscher's theory. For small $\ensuremath{\delta}$ we find a very good agreement for $l\ensuremath{\gtrsim}4$ or 7 depending on the excitation considered.
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